Practice Problems :: Simplex Method

Problem 1:

Consider the following feasible set:

  • x \ge -1 \text{ and } x\le 1
  • y \ge -1 \text{ and } y\le 1
  • z \ge -1 \text{ and } z\le 1
  • w \ge -1 \text{ and } w\le 1

The vertices of this feasible set have four coordinates, and each coordinate is either equal to positive one or negative one. For example (1,-1,1,1) is a vertex of the feasible set.

Two vertices are neighbors if they agree in all but one coordinate. For example, (1,1,1,1) is a neighbor of (1,-1,1,1).

Starting with the vertex (1,-1,1,1) as v_0, maximize the following objective function: -2x-y+3z+4w

Problem 2:

The permutahedron, P_nis an example of a feasible set. The vertices of the permutahedron are given by permutations of n (we’ll talk more about these in the coming weeks). I have listed the vertices of the permutahedron P_3 below:

  • (1,2,3)
  • (2,1,3)
  • (1,3,2)
  • (2,3,1)
  • (3,1,2)
  • (3,2,1)

Correction:

The neighbors a vertex are obtained by swapping entries that are adjacent in value.

  • For example: (1,2,3) and (2,1,3) are neighbors. But (3,1,2) and (2,1,3) are also neighbors. In the second example, I am looking at the entries whose values are 3 and 2, and swap.

Problem 2 part 1: Show that each vertex of P_3 has exactly two neighbors. (Thus, this feasible set is a polygon, even though it naturally lives in 3-space.)

Problem: 2 part 2: Use the simplex method, starting at the vertex (1,2,3) to maximize the objective function 2x+4y+z